Mass and stiffness asymmetry in rotating machinery is often the manifestation of developing imperfections or faults. Any asymmetry in a rotating part adds a periodic term to the coefficients of the equations of motion with a magnitude proportional to the level of asymmetry and frequency of twice the rotation speed. There are several factors affecting the response at the speed of rotation and its multiples, therefore asymmetry has a negligibly small effect on the response magnitude (vibration). At these frequencies, asymmetry is very difficult to detect from the naturally arising response at an early stage. In theory, many faults do produce special features in the response in the form of additional harmonics or modulation, but in practice, due to the small magnitude of these signal components compared with the unbalance and normal dynamic response, faults are only detected when they are severe.
Asymmetry of a rotating system may arise from design specifications, fulfilling certain engineering requirements (e.g. wind turbines and impellers), or could be the result of developing faults such as shaft crack, case rubbing or turbine blade failure. In the latter, a dedicated diagnostic procedure of the rotating structure is required to monitor the machine, as an aid to schedule maintenance shut-downs, only when necessary. Mathematically, asymmetric rotating systems are characterized by periodic (usually harmonic) coefficients which may appear in the mass, damping or stiffness matrices. These periodic terms, can cause a parametric resonance or give rise to self excited vibrations.
Analytical and numerical tools for investigating time varying systems include the Lyapunov-Floquet transformation whereby a transition matrix transforms any Linear Time Periodic (LTP) equation of motion to a linear time invariant (LTI) differential equation.
In some rotating linear time variant systems, the Floquet-Lyapunov transformation may take the form of a coordinate system's transformation which results in constant coefficient matrices. The transformation matrix is obtained by defining two coordinate systems, inertial and rotating (body fixed).
Most of the publications related to rotordynamics focus on critical speeds prediction balancing procedures and stability analysis.
External excitation devices (e.g. magnetic bearings) open new possibilities for active detection of faults. The main source of excitation (misalignment and unbalance) in rotating machines appears at the multiples of the speed of rotation. It is often the case that a measurement of the normal response cannot be separated from the effect of a developing fault since minute defects are buried under larger signal components appearing at the same frequencies. Active diagnostics, on the other hand, is capable of injecting a dedicated interrogation force at non-synchronous excitation frequencies and with the help of appropriate models of the rotating system, a unique signal frequency that is the result of a specific defect may be created. With this model-based diagnostics, trending is no longer necessary and superior detectability of certain faults can be achieved. The appearance of tiny additional spectral lines in the presence of cracks was observed by Bucher and Seibold (“A two-stage approach for enhanced diagnosis of rotating machines”, IFToM Conference on Rotor Dynamics, Darmstadt, Germany, pp 338-349, September 1998), in which both a model based and a signal based detection approach was proposed.
The present invention seeks to exploit the advantages of active diagnostics for the detection of asymmetry in the rotating part, by utilizing an external excitation device (e.g. Active Magnetic Bearing (AMB)) as a non-synchronous force exciter. The steady state response, expressed in inertial coordinates is shown to incorporate side-bands. While the carrier frequency is largely related to the symmetric part of the system, the magnitude of side-band frequency lines is associated with the level of asymmetry in the system. The concept and implications of non-synchronous excitation on the measured response are discussed hereinafter, with reference to several models, for the more simplified of which, analytical steady-state solutions are obtained. The rigid model is used to reveal the inherent ability to actively detect asymmetry. Flexible shaft models, in which the shaft's mass and gyroscopic effects are neglected, are then derived and solved to prove the advantages of the non-synchronous excitation over the synchronous excitation scheme. Detecting rotating asymmetry in the presence of anisotropic stator is demonstrated through an approximated solution using Hill's infinite determinant.
We further investigate the gyroscopic effect on the asymmetry detection. Using the Campbell diagram and the frequency response at different rotation speeds, a working point (rotation and excitation frequencies) is chosen, such that the modulated response indicative to the asymmetry would resonate. Further, a more realistic model in which finite element formulation is used and the shaft's mass is taken into account. The detection of asymmetry by means of non-synchronous excitation is demonstrated in the presence of a combined synchronous and asynchronous excitation sources.